5.2.8 Parametrized Covariant Transport
Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. To every morphism $f: C \rightarrow D$ in the $\infty $-category $\operatorname{\mathcal{C}}$, Definition 5.2.2.4 associates a covariant transport functor $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$, which is uniquely determined up to isomorphism (see Proposition 5.2.2.8). Our goal in this section is to show that the functor $f_{!}$ can be chosen to depend functorially on the morphism $f$: that is, the construction $f \mapsto f_{!}$ can be promoted to a functor from the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ to the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )$. We begin by introducing a more elaborate version of Definition 5.2.2.4.
Definition 5.2.8.1 (Parametrized Covariant Transport). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $C$ and $D$ be vertices of $\operatorname{\mathcal{C}}$. We will say that a morphism $F: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_ C \rightarrow \operatorname{\mathcal{E}}_{D}$ is given by parametrized covariant transport if there exists a morphism of simplicial sets $\widetilde{F}: \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_ C \rightarrow \operatorname{\mathcal{E}}$ satisfying the following conditions:
- $(1)$
The diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \ar [r]^-{ \widetilde{F} } \ar [d] & \operatorname{\mathcal{E}}\ar [d]^-{U} \\ \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \ar [r] & \operatorname{\mathcal{C}}} \]
commutes (where the lower horizontal map is induced by the inclusion $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \hookrightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$).
- $(2)$
The restriction $\widetilde{F}|_{ \{ 0\} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C}}$ is given by projection onto $\operatorname{\mathcal{E}}_{C}$, and the restriction $\widetilde{F}|_{ \{ 1\} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} }$ is equal to $F$.
- $(3)$
For every edge $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$ and every object $X \in \operatorname{\mathcal{E}}_{C}$, the composite map
\[ \Delta ^1 \times \{ f\} \times \{ X\} \hookrightarrow \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_ C \xrightarrow { \widetilde{F} } \operatorname{\mathcal{E}} \]
is a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.
If these conditions are satisfied, we say that the morphism $\widetilde{F}$ witnesses $F$ as given by parametrized covariant transport.
Example 5.2.8.3. Let $\operatorname{Set}_{\ast }$ denote the category of pointed sets (Example 4.2.3.3), and let $V: \operatorname{Set}_{\ast } \rightarrow \operatorname{Set}$ denote the forgetful functor $(X,x) \mapsto X$. Then the induced map $\operatorname{N}_{\bullet }(V): \operatorname{N}_{\bullet }( \operatorname{Set}_{\ast } ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set})$ is a cocartesian fibration (in fact, it is a left covering map), whose fiber over an object $X \in \operatorname{N}_{\bullet }(\operatorname{Set})$ can be identified with the set $X$. For every pair of sets $X$ and $Y$, the evaluation map
\[ \operatorname{ev}: \operatorname{Hom}_{\operatorname{Set}}(X,Y) \times X \rightarrow Y \quad \quad (f,x) \mapsto f(x) \]
is given by parametrized covariant transport (in the sense of Definition 5.2.8.1).
Proposition 5.2.8.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and let $C$ and $D$ be vertices of $\operatorname{\mathcal{C}}$. Then:
There exists a morphism $F: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ which is given by parametrized covariant transport.
An arbitrary diagram $F': \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ is given by parametrized covariant transport if and only if it is isomorphic to $F$ (as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )$).
Proof.
Apply Lemma 5.2.2.13 to the simplicial set $K = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C}$.
$\square$
Variant 5.2.8.6 (Parametrized Contravariant Transport). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets and let $C$ and $D$ be vertices of $\operatorname{\mathcal{C}}$. Applying Proposition 5.2.8.4 to the opposite cocartesian fibration $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$, we obtain a diagram $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{D}, \operatorname{\mathcal{E}}_{C} )$, carrying each edge $f: C \rightarrow D$ to a functor $f^{\ast }: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ given by contravariant transport along $f$.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Recall that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ can be identified with the fiber over $Y$ of the left fibration $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ of Proposition 4.6.4.11, or with the fiber over $X$ of the right fibration $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} $. In either case, parametrized transport recovers the composition law of $\operatorname{\mathcal{C}}$:
Proposition 5.2.8.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $C$, $D$, and $E$. Then the composition law
\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E) \]
of Construction 4.6.9.9 is given by parametrized covariant transport for the left fibration $U: \{ C\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (in the sense of Definition 5.2.8.1), and also by parametrized contravariant transport for the right fibration $V: \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ E\} \rightarrow \operatorname{\mathcal{C}}$.
Proof.
We will prove the first assertion; the second follows by a similar argument. Let $S: \Delta ^1 \times \Delta ^1 \rightarrow \Delta ^2$ be the morphism given on vertices by the formula $T(i,j) = i(j+1)$, and let $T$ be a section of the trivial Kan fibration $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ (see Corollary 4.6.9.5). Then the composite map
\[ \Delta ^1 \times \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \xrightarrow {S \times T} \Delta ^2 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \rightarrow \operatorname{\mathcal{C}} \]
carries $\{ 0\} \times \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ to the vertex $C$, and can therefore be identified with a functor
\[ \widetilde{F}: \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \{ C\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}. \]
which exhibits the composition law as given by parametrized covariant transport for the left fibration $U$.
$\square$
Proposition 5.2.5.1 has a counterpart for parametrized covariant transport:
Proposition 5.2.8.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. Let $C$, $D$, and $E$ be objects of $\operatorname{\mathcal{C}}$, and let
\[ F: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D} \quad \quad G: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{E} \]
\[ H: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{E} \]
be given by parametrized covariant transport. Then the diagram
5.25
\begin{equation} \begin{gathered}\label{equation:transitivity-parametrized} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \ar [r]^-{ \operatorname{id}\times F } \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{\mathcal{E}}_{D} \ar [d]^{ G } \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E) \times \operatorname{\mathcal{E}}_{C} \ar [r]^-{ H } & \operatorname{\mathcal{E}}_{E} } \end{gathered} \end{equation}
commutes in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$; here the left vertical map is given by the composition law of Construction 4.6.9.9.
Proof.
Let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E)$ be the Kan complex defined in Notation 4.6.9.1, let $H'$ denote the composite map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E) \times \operatorname{\mathcal{E}}_{C} \xrightarrow {H} \operatorname{\mathcal{E}}_{E}, \]
and let $H''$ denote the composition
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} & \rightarrow & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \\ & \xrightarrow {F} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{\mathcal{E}}_{D} \\ & \xrightarrow {G} & \operatorname{\mathcal{E}}_{E}. \end{eqnarray*}
We will show that $H'$ and $H''$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{E} )$. The homotopy commutativity of the diagram (5.25) will then follow by precomposing with any section of the trivial Kan fibration $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$.
Choose morphisms
\[ \widetilde{F}: \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}} \]
\[ \widetilde{G}: \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}} \]
\[ \widetilde{H}: \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}} \]
which witness $F$, $G$, and $H$ as given by parametrized covariant transport, respectively. Composing with the projection maps
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \leftarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E), \]
we obtain morphisms
\[ \widetilde{F}': \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}} \]
\[ \widetilde{H}': \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}. \]
Let $\widetilde{G}'$ denote the composite map
\begin{eqnarray*} \operatorname{N}_{\bullet }( \{ 1 < 2\} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} \hspace{-.5em} & \rightarrow & \operatorname{N}_{\bullet }( \{ 1 < 2\} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \\ & \xrightarrow { F } & \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{\mathcal{E}}_{D} \\ & \xrightarrow { \widetilde{G} } & \operatorname{\mathcal{E}}. \end{eqnarray*}
Since $U$ is an inner fibration, the lifting problem
\[ \xymatrix@C =100pt@R=50pt{ \Lambda ^{2}_{1} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} \ar [r]^-{ ( \widetilde{G}', \bullet , \widetilde{F}' ) } \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \Delta ^2 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} \ar [r] \ar@ {-->}[ur]^{\Phi } & \operatorname{\mathcal{C}}} \]
admits a solution $\Phi : \Delta ^2 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}$. Let $\widetilde{H}''$ denote the restriction of $\Phi $ to the product $\operatorname{N}_{\bullet }( \{ 0, 2\} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C}$. Using Proposition 5.1.4.12, we see that $\widetilde{H}''$ is a $U$-cocartesian lift of $U \circ \widetilde{H}'' = U \circ \widetilde{H}'$, in the sense of Definition 5.2.2.10. Applying the uniqueness assertion of Lemma 5.2.2.13, we conclude that the restrictions $H' = \widetilde{H}'|_{ \{ 2\} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} }$ and $H'' = \widetilde{H}''|_{ \{ 2\} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} }$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{E})$, as desired.
$\square$
Using Proposition 5.2.8.8, we obtain the following refinement of Construction 5.2.5.2:
Construction 5.2.8.9 (Enriched Homotopy Transport: Covariant Case). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories and let us regard the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as enriched over the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Kan complexes (Construction 4.6.9.13). It follows from Proposition 5.2.8.8 (and Example 5.2.2.5) that there is a unique $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ with the following properties:
For each object $C$ of the $\infty $-category $\operatorname{\mathcal{C}}$, $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ is the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ (regarded as an object of $\mathrm{h} \mathit{\operatorname{QCat}}$).
For every pair of objects $C,D \in \operatorname{\mathcal{C}}$, the induced map
\[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )^{\simeq } \]
in $\mathrm{h} \mathit{\operatorname{Kan}}$ corresponds to the parametrized covariant transport functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ of supplied by Proposition 5.2.8.4 (which is well-defined up to isomorphism).
We will refer to $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the enriched homotopy transport representation of the cocartesian fibration $U$. Note that the underlying functor of ordinary categories $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ coincides with homotopy transport representation of Construction 5.2.5.2.
Variant 5.2.8.11 (Enriched Homotopy Transport: Left Fibrations). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories. Applying Construction 5.2.8.9, we obtain an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor
\[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}, \]
given on objects by the formula $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}( C ) = \{ C \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.
Variant 5.2.8.12 (Enriched Homotopy Transport: Contravariant Case). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of $\infty $-categories. Applying Construction 5.2.8.9 to the opposite functor $U^{\operatorname{op}}$, we deduce that there is a unique $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ with the following properties:
For each object $C$ of the $\infty $-category $\operatorname{\mathcal{C}}$, $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ is the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ (regarded as an object of $\mathrm{h} \mathit{\operatorname{QCat}}$).
For every pair of objects $C,D \in \operatorname{\mathcal{C}}$, the induced map
\[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{D}, \operatorname{\mathcal{E}}_{C} )^{\simeq } \]
is given by the parametrized contravariant transport functor $\operatorname{\mathcal{E}}_{D} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{\mathcal{E}}_{C}$ of Variant 5.2.8.6.
We will refer to $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the enriched homotopy transport representation of the cartesian fibration $U$. If $U$ is a right fibration, then $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ takes values in the full subcategory $\mathrm{h} \mathit{\operatorname{Kan}} \subseteq \mathrm{h} \mathit{\operatorname{QCat}}$.
Example 5.2.8.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category, which we regard as enriched over the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Kan complexes. Applying Proposition 5.2.8.7, we obtain the following:
For every object $C \in \operatorname{\mathcal{C}}$, the corepresentable $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor
\[ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \]
is the enriched homotopy transport representation for the left fibration $\{ C\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$.
For every object $D \in \operatorname{\mathcal{C}}$, the representable $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor
\[ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \]
is the enriched homotopy transport representation for the right fibration $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ D\} \rightarrow \operatorname{\mathcal{C}}$.